The Hartogs extension theorem on (n – 1)-complete complex spaces

Merker, Joël; Porten, Egmont

doi:10.1515/crelle.2009.088

Cited by 19 publications

(18 citation statements)

References 23 publications

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“…Let us point out that Theorem 1.2 holds for forms with weaker than L 2 -regularity and for complex spaces with arbitrary singularities. As a corollary of Theorems 1.1 and 1.2 we can obtain an analytic proof of the following theorem of Merker and Porten: Merker and Porten prove in [16] an extension theorem for meromorphic functions as well. It is not clear to us at the moment how to employ ∂-techniques to attack the extension problem for such functions.…”

Section: Introductionmentioning

confidence: 84%

Semiglobal results for $\overline∂$ on complex spaces with arbitrary singularities, Part II

Øvrelid

Vassiliadou

2011

Trans. Amer. Math. Soc.

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Abstract. We obtain some L 2 -results for ∂ on forms that vanish to high order on the singular set of a complex space. As a consequence of our main theorem we obtain weighted L 2 -solvability results for compactly supported ∂-closed (p, q)-forms (0 ≤ p ≤ n, 1 ≤ q < n) on relatively compact subdomains Ω of the complex space that satisfy H n−q (Ω, S) = 0 = H n−q+1 (Ω, S) for every coherent O X -module S. The latter result can be used to give an alternate proof of a theorem of Merker and Porten.

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Section: Introductionmentioning

confidence: 84%

Semiglobal results for $\overline∂$ on complex spaces with arbitrary singularities, Part II

Øvrelid

Vassiliadou

2011

Trans. Amer. Math. Soc.

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“…See [18] for a further discussion. For related results proved by other methods see, e.g., [16,22,21].…”

Section: )mentioning

confidence: 94%

Weighted Koppelman formulas and the ∂¯-equation on an analytic space

Andersson

Samuelsson

2011

Journal of Functional Analysis

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“…The first one is the following rigidity property: every analytic chaotic Levi-flat hypersurface is tangent to a singular complex algebraic foliation defined on the ambient surface. This fact follows from extension techniques of analytic objects in modified Stein spaces, which are now classic, see [22,25,26,32], but that we detail in this text, particularly the delicate passage through critical levels of the plurisubharmonic exhaustion function. Thus, chaotic Levi-flat hypersurfaces appear as regular invariant sets of algebraic differential equations.…”

Section: Annales De L'institut Fouriermentioning

confidence: 98%

Levi-flat hypersurfaces and their complement in complex surfaces

Gonzalez¹

2017

Ann. inst. Fourier

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In this work we study analytic Levi-flat hypersurfaces in complex algebraic surfaces. First, we show that if this foliation admits chaotic dynamics (i.e. if it does not admit a transverse invariant measure), then the connected components of the complement of the hypersurface are modifications of Stein domains. This allows us to extend the CR foliation to a singular algebraic foliation on the ambient complex surface. We apply this result to prove, by contradiction, that analytic Levi-flat hypersurfaces admitting a transverse affine structure in a complex algebraic surface have a transverse invariant measure. This leads us to conjecture that Levi-flat hypersurfaces in complex algebraic surfaces that are diffeomorphic to a hyperbolic torus bundle over the circle are fibrations by algebraic curves.

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The Hartogs extension theorem on (n – 1)-complete complex spaces

Cited by 19 publications

References 23 publications

Semiglobal results for $\overline∂$ on complex spaces with arbitrary singularities, Part II

Semiglobal results for $\overline∂$ on complex spaces with arbitrary singularities, Part II

Weighted Koppelman formulas and the ∂¯-equation on an analytic space

Levi-flat hypersurfaces and their complement in complex surfaces

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